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Lecture 3: Field Limitations
and Superconductivity
Dr G Burt
Lancaster University
Engineering
Average Heating
• In normal conducting cavities, the RF deposits large
amounts of power as heat in the cavity walls.
• This heat is removed by flushing cooling water through
special copper cooling channels in the cavity. The faster
the water flows (and the cooler), the more heat is
removed.
• For CW cavities, the cavity temperature reaches steady
state when the water cooling removes as much power as
is deposited in the RF structure. (Limit is ~ 1 MW but 500
kW is safer)
• This usually is required to be calculated in a Finite
Element code to determine temperature rises.
• Temperature rises can cause surface deformation,
surface cracking, outgassing or even melting.
• By pulsing the RF we can reach much higher gradients
as the average power flow is much less than the peak
power flow.
Pulsed Heating
The power deposited is
Rs H max 2
Pd 
2
0
Rs 
2
And the temperature on the surface
is
T 
Pulsed RF however has problems due to heat
diffusion effects.
Over short timescales (<10ms) the heat doesn’t
diffuse far enough into the material to reach the
water cooling. This means that all the heat is
deposited in a small volume with no cooling.
Cyclic heating can lead to surface damage if the
temperature rise creates thermal stress (~40 K) .
Pd t pulse
2  ce
 is density,  is thermal conductivity and
ce is specific heat
Gas Breakdown
• If we apply a high
voltage across a
gap we can ionise
the molecules in the
intervening gas.
• At high pressure the
mean free path is
too low to gain
enough momentum
• At low pressure
there are not
enough molecules
to ionise.
Does this mean we don’t get breakdown in
vacuum?
Field Emission
• High electric fields can lead to electrons quantum
tunnelling out of the structure creating a field emitted
current.
Once emitted this field emitted
current can interact with the
cavity fields.
Although initially low energy, the
electrons can potentially be
accelerated to close to the
speed of light with the main
electron beam, if the fields are
high enough.
This is known as dark current
trapping.
Field Enhancement
• The surface of an accelerating
structure will have a number of
imperfections at the surface
caused by grain boundaries,
scratches, bumps etc.
• As the surface is an
equipotential the electric fields
at these small imperfections
can be greatly enhanced.
• In some cases the field can be
increase by a factor of several
hundred.
10000
Beta
Elocal=b E0
100000
1000
100
10
2b
1
1
10
100
h/b
h
1000
Vacuum Breakdown
• Breakdown occurs when a
plasma discharge is generated in
the cavity.
• This is almost always associated
with some of the cavity walls
being heated until it vaporises
and the gas is then ionised by
field emission. The exact
mechanisms are still not well
understood.
• When this occurs all the incoming
RF is reflected back up the
coupler.
• This is the major limitation to
gradient in most pulsed RF
cavities and can permanently
damage the structure.
Kilpatrick Limits
• A rough empirical formula for the peak surface electric
field is
• It is not clear why the field strength decreases with
frequency.
• It is also noted that breakdown is mitigated slightly by
going to lower group velocity structures.
• The maximum field strength also varies with pulse length
as t-0.25 (only true for a limited number of pulse lengths)
• As a SCRF cavity would quench long before breakdown,
we only see breakdown in normal conducting structures.
Dark Current Trapping
• When we looked at beam dynamics we saw
that we could inject a low energy bunch in a
beta=v/c=1 structure and it could be
accelerated to the speed of light and arrive
on crest.
• If we have field emitted electrons in the
structure these could also be capture and can
travel with the main beam.
• The gradient at which this occurs is given by
Maximum Gradient Limits
• All the limiting
factors scale
differently with
frequency.
• They also mostly
vary with pulse
length.
• The limiting
factor tends to
be different from
cavity to cavity.
For a CW machine the gradient is limited by average heating instead. Also
need to think about the electricity bill as 1 MW is £200 per day.
SCRF Cavities
The power required to keep a
cavity on filled to a set voltage is
the power extracted by the beam
plus the ohmic heating in the
walls.
In order to increase the efficiency
of coupling power to the beam we
need to minimise the ohmic
heating in the cavity walls.
The ohmic heating can be reduced by 5-6 orders of magnitude with the
use of a superconducting cavity.
DC Superconductivity
• Electrons are fermions and obey the Pauli
exclusion principle.
• In some materials, below a certain temperature,
the motion of electrons distort the lattice.
• This distortion can allow a small net attraction
between two electrons.
• The electrons then condense into pairs, known
as cooper pairs.
• Cooper pairs are bosons and can all occupy the
same state.
• Hence the cooper pairs can flow without being
scattered.
Miessner Effect
• Normal Conductor
H>0
T > Tc
H>0
T < Tc
H=0
T < Tc
H>0
T > Tc
H>0
T < Tc
H=0
T < Tc
• Superconductor
RF Superconductivity
• When a DC electric field is applied to a
superconductor all the current is
carried by the cooper pairs, which flow
without resistance to shield the field
inside the material.
• If an RF field is applied the same thing
occurs except when the field switches
direction, although the pairs have no
friction, they do have inertia.
• The inertia means that the fields are
not screened perfectly and the
penetrating field can interact with the
normal electrons causing a small
resistance.
RF superconductivity
The surface resistance has
the following dependence
• Rs increases with frequency squared
• Rs increases exponentially with
temperature
SCRF cavities have higher losses
as they increase in frequency,
for this reason there are few
SCRF cavities above 4GHz.
2
RBCS
1 f 
 17.67 
 2 10
  exp  

T  1.5 
T


4
Residual Resistance
Rres is normally
between 1-10 Ohms
An SRF cavity will decrease its resistance with temperature in theory.
However in practice there is often a minimum resistance due to the effects of
normal conducting impurities in the niobium.
One of the main effects is flux pinning where magnetic fields are frozen into
normal conducting impurities inside the superconductor. This can be avoided by
shielding the cavity from magnetic fields during cooldown.
RF Critical B field
When the electrons condence into
cooper pairs the resulting
superconducting state is more
ordered than the normal-conducting
state.
When a magnetic field is applied to a superconductor, supercurrents flow.
This increases the free energy of the superconducting state. When the free
energy of the superconducting state equals the normal conducting state the
flux enters the material.
For RF fields the flux continues to be excluded in a metastable state unit the
field reaches the critical superheating field (240 mT)
Field Emission
Once emitted a field emitted current can
interact with the cavity fields.
Although initially low energy, the electrons can
be accelerated by the RF fields and deposit
their energy on the cavity walls, hence
heating them.
Limits operation from most cavities between
20-35 MV/m (depending on cleaning
processes and luck).
SRF Couplers
• Also a limited power in SRF couplers.
• WG limited to 500 kW due to multipactor (electron
cloud).
• Coax is limited to a similar amount by limited cooling of
inner coax.
Multipactor
Multipactor is a resonant electron
instability where an electron striking
the surface with a given energy
generates statistically more than
one secondary particle.
If the secondaries are returned
to the surface at the same rf
phase and energy as the
primary then an exponential
growth in electrons will occur.
These electrons can absorb all
additional RF power from the
cavity keeping the cavity from
increasing it’s voltage.
Parallel Plate Multipactor
One of the most common type of multipactor is two-point
multipactor that occurs between two metallic plates.
The RF fields accelerate the particle across the gap. The RF fields
reverse every half period so the electron motion is resonant if it
take (n+1/2) periods to traverse the gap
a.c. electric field
across gap or in
waveguide
HPRF MSc
© Lancaster University
21
Secondary
SecondaryEmission
Electron Yield
• For multipactor to occur each electron striking the surface must
generate more than one secondary on average.
• The ratio of primary to secondary electrons is given by the
secondary emission yield (SEY).
• SEY is strongly
dependant on
impact energy
and only gives
an SEY>1 for a
finite range of
low energies
HPRF MSc
© Lancaster University
22
Multipactor in Cavities
Electron
trajectory
Two point multipactor can also occur in pillbox
and elliptical cavities.
The electron trajectories are bent by the RF
magnetic field to form semi-circles.
In pillbox cavities trajectories are resonant if
the cyclotron conditions are met
Bo 
m
e
Multipactor will occur if the electric field
accelerates the electrons to energies between
~50-500 eV depending on the material.
In elliptical cavities the resonant condition
varies with radius as smaller arcs are required
causing any resonant trajectories to break up,
preventing multipactor.
Microphonics
Microphonics: Changes in frequency
caused by connections to the outside
world
•Vibrations
•Pressure Fluctuations
This means the cavity is not always on
resonance and will require more RF
power to fill.
Additionally the constantly varying
frequency will cause phase errors.
To avoid problems we need to artificially
broaden the cavity bandwidth by using
a lower Qe of at least 106
Cryomodules
The cryovessel is
often one of the more
complicated items
required. It requires:
•Inlet and return pipes
for He and N.
•It must support the
cavity.
•Have low static heat
loss.
•Incorporate all
couplers.
Filling factor
• The cryomodule in
an SRF system also
takes up significant
space.
• The filling factor is
the ratio of the cavity
length to cryomodule
length.
• It varies from a factor
of 5 for single cells to
1.5 for a 9 cell cavity
(typically its 2 extra
cells on each side).
Cryogenic systems
• All refrigerators have a technical efficiency, ηT of 20%-30%
• The Carnot efficiency is given by
T
c 
300  T
• The dynamic heat load, Pc, is the rf power dissipated in the
cavity walls.
• A static heat load, Ps, adds an additional heating (~5-10 W)
• Liquid helium transfer lines require 1W per metre, so total
loss is length L (More efficient lines can be used)
• It is standard to fill to an overcapacity, O (~50%)
Total Power for N cryostats= O N (Pc +Ps +L) / (ηT ηc )
RF Cavities for Linacs and
Circular Accelerators
Circular Accelerators
Linac (High energy)
•High HOM damping
•CW operation  high power
couplers required
•High gradient cavities
•Multicell
•Pulsed operation (often)
CESR cavity
TESLA cavity
SCRF vs NCRF
SCRF
• More efficient (even
when including
cryogenic losses)
• Higher CW gradient
• Long pulse or CW only
• Complex system
needing cryostats and
cryogenics
• Only frequencies below
4 GHz.
NCRF
• Less efficient.
• Higher pulsed gradient
• Simpler systems, water
cooled
• More reliable
• Lower capital costs
• Smaller apertures mean
higher wakefields